Alternative convergence criteria for iterative methods of solving nonlinear equations

Document Type

Article

Publication Date

1-1-1984

Abstract

Let χm+1=T(χm) or even χm+1=T(χm,χm-1, ..., χm-q), m=1,2,3 ... be an iteration method for solving the nonlinear problem F(χ)=0, where F(χ) and its derivatives possess all of the properties required by T(χm). Then if it can be established that for the problem at hand ∥F(χm+1)∥≤βm∥F(χm)∥, ∀ m > M0 (M0<∞) and 0≤βm<1, definitions are established and theorems proven concerning convergence, uniqueness and bounds on the error after 'm' successive iterations of a new approach to convergence properties T(χm). These charateristics are referred to as "alternate" (local, global) convergence properties and none of the proofs given are restricted to any specific type of method such as, e.g. contraction mapping types. Application of results obtained are illustrated using Newton's method as well as the general concept of Newton-like methods. © 1984.

Identifier

0021372299 (Scopus)

Publication Title

Journal of the Franklin Institute

External Full Text Location

https://doi.org/10.1016/0016-0032(84)90035-8

ISSN

00160032

First Page

89

Last Page

103

Issue

2

Volume

317

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